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Numbers

Source: Canada Mathematical Olympiad 2007

August 1, 2007
algebrapolynomialratiotrigonometryinvariantinductioncombinatorics unsolved

Problem Statement

For two real numbers a a, b b, with ab≠1 ab\neq 1, define the ∗ \ast operation by a∗b=a+b−2ab1−ab. a\ast b=\frac{a+b-2ab}{1-ab}. Start with a list of n≥2 n\geq 2 real numbers whose entries x x all satisfy 0<x<1 0<x<1. Select any two numbers a a and b b in the list; remove them and put the number a∗b a\ast b at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. a. a. Prove that this single number is the same regardless of the choice of pair at each stage. b. b. Suppose that the condition on the numbers x x is weakened to 0<x≤1 0<x\leq 1. What happens if the list contains exactly one 1 1?
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